Communications of the
Korean Mathematical Society
CKMS

ISSN(Print) 1225-1763 ISSN(Online) 2234-3024

Article

HOME ALL ARTICLES View

Commun. Korean Math. Soc. 2020; 35(3): 745-757

Online first article March 6, 2020      Printed July 31, 2020

https://doi.org/10.4134/CKMS.c190364

Copyright © The Korean Mathematical Society.

Integer points on the elliptic curves induced by Diophantine triples

Jinseo Park

Catholic Kwandong University

Abstract

A set $\{a_1, a_2, \dots, a_m\}$ of positive integers is called a Diophantine $m$-tuple if $a_ia_j+1$ is a perfect square for all $1\leq i < j \leq m$. In this paper, we find the structure of a torsion group of elliptic curves $E_k$ constructed by a Diophantine triple $\{F_{2k}, F_{2k+2}, 4F_{2k+1}F_{2k+2}F_{2k+3}\}$, and find all integer points on the elliptic curve under assumption that rank$(E_k(\mathbb{Q}))=2$.

Keywords: Diophantine $m$-tuple, Fibonacci numbers, elliptic curve

MSC numbers: Primary 11B39, 11G05, 11D09; Secondary 11D45

Supported by: This work was supported by the National Research Foundation of Korea(NRF) grant funded by the Korea government(MSIT) (No. 2019R1G1A1006396)